Error Feedback Regulation Of One-Dimensional Anti-Stable Wave Equation

In control theory,the output regulation problem of partial differential equation(PDE)systems has been widely concerned and studied by experts and scholars.The anti-stable second-order elastic vibration system has a wide range of applications in reality because it can simulate many physical phenomena,such as acoustic instability.Therefore,many scientific researchers have devoted to its output regulation problems and have so far achieved fruitful results.In this paper,we mainly consider the one-dimensional(1-d)anti-stable wave equation with different types of distributed disturbances,and regulate the output of system to track the preset reference signal by error feedback.This thesis consists of four chapters.First chapter introduces the research background of the error feedback regulation problem(EFRP)of PDE and the existing classical methods,and give the related preliminary definitions and lemmas.In the second chapter,the EFRP of 1-d anti-stable wave equation with external disturbance is discussed.The harmonic disturbance and reference signal are both unknown.We first introduce a transport equation to deal with the anti-damper and obtain an equivalent auxiliary system by constructing a transformation.The distributed disturbance of the original system is transformed to the boundary and the output is changed into a multiple of the tracking error.Then,the tracking error signal and its derivative are used to design an adaptive state observer for the auxiliary system,which recover all the state information of the system and updates the unknown parameters.A feedback controller is proposed which transforms the regulation problem of the original system into a stabilization problem of the closedloop system.We select the appropriate state space and define the system operator.The stability of the closed-loop system is proved via the operator semigroup theory and the preset target tracking is achieved.Finally,some numerical simulations are given to verify the theoretical results.In the third chapter,we consider the EFRP of 1-d anti-stable wave equation with distributed disturbance.The disturbance and reference signal generated by a finite-dimensional external system are both unknown.We obtain an effective equivalent auxiliary system by constructing regulator equations.Then an infinitedimensional state observer is designed for the auxiliary system.It should be noted that there is no need to update the unknown parameters as in chapter 2.Next,an observer-based error feedback controller is proposed and the exponential stability of the closed-loop system is proved by using the operator semigroup theory.Finally,some numerical simulations are given to verify the theoretical results.The fourth chapter makes a summary of the full text and makes a prospect for the future work.